A solid cylinder of mass M and radius R is mounted to an axle through its center. The axle is attached to a horizontal spring of constant k, as shown in the figure. Initially the cylinder is at rest and the spring is un-stretched. The cylinder is then pulled a distance A and released. The cylinder rolls back and forth without slipping.
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
Assess: Assume that we have the same spring and mass system, only
now the cylinder is released from rest on a frictionless surface at a distance A
from the equilibrium position. What is the period of the simple harmonic motion
of this system? Express your answer in terms of k and M
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